The From omega to epsilon nought blog lists out all the ordinals between \(\omega\) to \(\epsilon_0\) in detail assuming n=3 when substituting finite values for \(\omega\). This blog summarises the detail and calculates the number of steps in the progression.
The progression is used to compare big numbers in the Strong D Function blog post.
Recap of all steps from omega to epsilon nought[]
1. add \(\omega\) 9 times to get \(\omega^{\omega}\)
2. add \(\omega^{\omega}\) 3 times to get \(\omega^{\omega+1}\)
3. add \(\omega^{\omega+1}\) 3 times to get \(\omega^{\omega+2}\)
4. add \(\omega^{\omega+2}\) 3 times to get \(\omega^{\omega.2}\)
5. add \(\omega^{\omega.2}\) 3 times to get \(\omega^{\omega.2+1}\)
6. add \(\omega^{\omega.2+1}\) 3 times to get \(\omega^{\omega.2+2}\)
7. add \(\omega^{\omega.2+2}\) 3 times to get \(\omega^{\omega^2}\)
8. add \(\omega^{\omega^2}\) 3 times to get \(\omega^{\omega^2+1}\)
9. add \(\omega^{\omega^2+1}\) 3 times to get \(\omega^{\omega^2+2}\)
10. add \(\omega^{\omega^2+2}\) 3 times to get \(\omega^{\omega^2+\omega}\)
11. add \(\omega^{\omega^2+\omega}\) 3 times to get \(\omega^{\omega^2+\omega+1}\)
12. add \(\omega^{\omega^2+\omega+1}\) 3 times to get \(\omega^{\omega^2+\omega+2}\)
13. add \(\omega^{\omega^2+\omega+2}\) 3 times to get \(\omega^{\omega^2+\omega.2}\)
14. add \(\omega^{\omega^2+\omega.2}\) 3 times to get \(\omega^{\omega^2+\omega.2+1}\)
15. add \(\omega^{\omega^2+\omega.2+1}\) 3 times to get \(\omega^{\omega^2+\omega.2+2}\)
16. add \(\omega^{\omega^2+\omega.2+2}\) 3 times to get \(\omega^{\omega^2.\omega.2+\omega}\)
17. add \(\omega^{\omega^2.2}\) 3 times to get \(\omega^{\omega^2.2+1}\)
18. add \(\omega^{\omega^2.2+1}\) 3 times to get \(\omega^{\omega^2.2+2}\)
19. add \(\omega^{\omega^2.2+2}\) 3 times to get \(\omega^{\omega^2.2+\omega}\)
20. add \(\omega^{\omega^2.2+\omega}\) 3 times to get \(\omega^{\omega^2.2+\omega+1}\)
21. add \(\omega^{\omega^2.2+\omega+1}\) 3 times to get \(\omega^{\omega^2.2+\omega+2}\)
22. add \(\omega^{\omega^2.2+\omega+2}\) 3 times to get \(\omega^{\omega^2.2+\omega.2}\)
23. add \(\omega^{\omega^2.2+\omega.2}\) 3 times to get \(\omega^{\omega^2.2+\omega.2+1}\)
24. add \(\omega^{\omega^2.2+\omega.2+1}\) 3 times to get \(\omega^{\omega^2.2+\omega.2+2}\)
25. add \(\omega^{\omega^2.2+\omega.2+2}\) 3 times to get \(\epsilon_0\)
Calculating the total number of steps[]
All of the above steps are recursive. For example, Step 8 requires adding \(\omega^{\omega^2}\) 3 times. But adding \(\omega^{\omega^2}\) once requires all of Step 7 (adding \(\omega^{\omega.2+2}\) 3 times) and so on. This continues all the way to Step 1 and adding \(\omega\) 9 times.
Therefore the number of steps in total is \(3^24.9 = 3^{26} = 3^{3^3-1}\)
The general rule for any n other than 3 is \(n^{n\uparrow\uparrow(n-1)-1}\)